Fractional-order differential equations (FDEs) excel at modelling systems with long-range memory, but many such systems also exhibit stochastic behaviour. While existing neural differential equation models can capture either memory effects (Neural FDEs) or stochastic dynamics (Neural SDEs), a unified framework efficiently combining both remains absent. We introduce the Neural Fractional Stochastic Differential Equation (Neural FSDE), a model that learns continuous-time dynamics with both memory and randomness from data. To enable scalable training, we derive an adjoint sensitivity method for Neural FSDEs. We demonstrate its practical utility on an option pricing task, where our Neural FSDE not only achieves superior accuracy compared to strong baselines like rough volatility models and standard Neural SDEs, but also offers an order-of-magnitude speedup in calibration time.